Algebra Examples

$\frac{2}{x + 3} + \frac{x + 3}{8 x - 1}$

Step 1

To write $\frac{2}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{8 x - 1}{8 x - 1}$ .

$\frac{2}{x + 3} \cdot \frac{8 x - 1}{8 x - 1} + \frac{x + 3}{8 x - 1}$

Step 2

To write $\frac{x + 3}{8 x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{2}{x + 3} \cdot \frac{8 x - 1}{8 x - 1} + \frac{x + 3}{8 x - 1} \cdot \frac{x + 3}{x + 3}$

Step 3

Write each expression with a common denominator of $(x + 3) (8 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{x + 3}$ by $\frac{8 x - 1}{8 x - 1}$ .

$\frac{2 (8 x - 1)}{(x + 3) (8 x - 1)} + \frac{x + 3}{8 x - 1} \cdot \frac{x + 3}{x + 3}$

Step 3.2

Multiply $\frac{x + 3}{8 x - 1}$ by $\frac{x + 3}{x + 3}$ .

$\frac{2 (8 x - 1)}{(x + 3) (8 x - 1)} + \frac{(x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 3.3

Reorder the factors of $(x + 3) (8 x - 1)$ .

$\frac{2 (8 x - 1)}{(8 x - 1) (x + 3)} + \frac{(x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 (8 x - 1) + (x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{2 (8 x) + 2 \cdot - 1 + (x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 5.2

Multiply $8$ by $2$ .

$\frac{16 x + 2 \cdot - 1 + (x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 5.3

Multiply $2$ by $- 1$ .

$\frac{16 x - 2 + (x + 3) (x + 3)}{(8 x - 1) (x + 3)}$

Step 5.4

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

$\frac{16 x - 2 + x (x + 3) + 3 (x + 3)}{(8 x - 1) (x + 3)}$

Step 5.4.2

Apply the distributive property.

$\frac{16 x - 2 + x \cdot x + x \cdot 3 + 3 (x + 3)}{(8 x - 1) (x + 3)}$

Step 5.4.3

Apply the distributive property.

$\frac{16 x - 2 + x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3}{(8 x - 1) (x + 3)}$

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $x$ by $x$ .

$\frac{16 x - 2 + x^{2} + x \cdot 3 + 3 x + 3 \cdot 3}{(8 x - 1) (x + 3)}$

Step 5.5.1.2

Move $3$ to the left of $x$ .

$\frac{16 x - 2 + x^{2} + 3 \cdot x + 3 x + 3 \cdot 3}{(8 x - 1) (x + 3)}$

Step 5.5.1.3

Multiply $3$ by $3$ .

$\frac{16 x - 2 + x^{2} + 3 x + 3 x + 9}{(8 x - 1) (x + 3)}$

Step 5.5.2

Add $3 x$ and $3 x$ .

$\frac{16 x - 2 + x^{2} + 6 x + 9}{(8 x - 1) (x + 3)}$

Step 5.6

Add $16 x$ and $6 x$ .

$\frac{22 x - 2 + x^{2} + 9}{(8 x - 1) (x + 3)}$

Step 5.7

Add $- 2$ and $9$ .

$\frac{22 x + x^{2} + 7}{(8 x - 1) (x + 3)}$

Step 5.8

Reorder terms.

$\frac{x^{2} + 22 x + 7}{(8 x - 1) (x + 3)}$